Time-resolved phosphorescence anisotropy (TPA) is used to measure the short-time rotational diffusion coefficient D-s(r)(phi) of charged tracer spheres as a function of the volume fraction phi of like-charged colloidal host spheres in nonaqueous solvents. Sphere interactions are varied from long-range repulsive to short-range attractive by changing the ionic strength and the solvent composition. It is shown that D-s(r)(phi) is very sensitive to details of the interaction near contact, in agreement with theory. In contrast, the low-shear viscosity eta(L)(phi) of the host dispersions is mostly controlled by the tail of the interaction potential. We discuss the applicability of Stokes-Einstein-Debye scaling D-s(r)(phi)proportional to1/eta(L)(phi), and D-s(r)(phi)proportional to1/eta(infinity)(phi), where eta(infinity) is the high-frequency-limiting viscosity. Scaling with eta(L)(phi) fails at high particle and low salt concentrations, while scaling with eta(infinity) is fairly good, in particular when an apparent nonstick boundary condition is imposed on the friction factor. We conclude that TPA is well suited for use as a microrheological technique.